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Theorem elpr 3518
Description: A member of an unordered pair of classes is one or the other of them. Exercise 1 of [TakeutiZaring] p. 15. (Contributed by NM, 13-Sep-1995.)
Hypothesis
Ref Expression
elpr.1  |-  A  e. 
_V
Assertion
Ref Expression
elpr  |-  ( A  e.  { B ,  C }  <->  ( A  =  B  \/  A  =  C ) )

Proof of Theorem elpr
StepHypRef Expression
1 elpr.1 . 2  |-  A  e. 
_V
2 elprg 3517 . 2  |-  ( A  e.  _V  ->  ( A  e.  { B ,  C }  <->  ( A  =  B  \/  A  =  C ) ) )
31, 2ax-mp 5 1  |-  ( A  e.  { B ,  C }  <->  ( A  =  B  \/  A  =  C ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 104    \/ wo 682    = wceq 1316    e. wcel 1465   _Vcvv 2660   {cpr 3498
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-v 2662  df-un 3045  df-sn 3503  df-pr 3504
This theorem is referenced by:  prmg  3614  difprsnss  3628  preqr1  3665  preq12b  3667  prel12  3668  pwprss  3702  pwtpss  3703  unipr  3720  intpr  3773  zfpair2  4102  elop  4123  ordtri2or2exmidlem  4411  onsucelsucexmidlem  4414  en2lp  4439  reg3exmidlemwe  4463  xpsspw  4621  acexmidlem2  5739  2oconcl  6304  exmidpw  6770  renfdisj  7792  fzpr  9825  maxabslemval  10948  xrmaxiflemval  10987  isprm2  11725  bj-zfpair2  13035  ss1oel2o  13116
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